1 Multilevel polynomial regression analysis

Multilevel polynomial regression analysis [MPRA; see Nestler et al. (2019)] is an adaptation of polynomial regression analysis [PRA; see Edwards & Parry (1993)] that can be used when data have a multilevel structure (e.g., observations across multiple time points nested within participants). In the current study, MPRA allows testing and interpreting both the linear effects of vitality and learning separately, as well as the effects of congruent vitality and learning scores using data collected at four time points with a one-month time lag between the assessments. In the following, we use the index m to denote the Level 2 unit (\(m = 1…N\)) and the index \(i\) to denote the Level 1 unit (\(i = 1, 2, …, n_{m}\)). To assess the existence of congruence effects, the following parameters are estimated at the population level: the slope of the line of congruence (LOC) (i.e., vitality = learning) as related to the outcome variable is given by \(\alpha_{1m} = b_{1} + b_{2}\). Here, \(b_{1}\) is the unstandardized regression coefficient for the (latent) vitality variable and \(b_2\) is the regression coefficient for the (latent) learning variable. The curvature along the line of perfect agreement is specified by \(\alpha_{1m} = b_{3} + b_{4} + b_{5}\). Here, \(b_{3}\) is the regression coefficient for (latent) vitality squared, b4 is the regression coefficient for the cross-product of the (latent) vitality and the (latent) learning variable, and b5 is the unstandardized regression coefficient for latent learning squared. The slope of the line of incongruence (LOIC) is defined as \(\alpha_{3m} = b_{1} - b_{2}\). The curvature of the line of incongruent vitality and learning levels as related to the outcome variable is specified as \(\alpha_{4m} = b_{3} - b_{4} + b_{5}\). Each of the regression coefficients \(b_{1}\) to \(b_{5}\) includes a fixed effect and Level 2 residual terms. The fixed effects of the regression coefficients are used to estimate the average response surface parameters \(\hat{\alpha}_{1}\) to \(\hat{\alpha}_{5}\) (Nestler et al., 2019).

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References

Edwards, J. R., & Parry, M. E. (1993). On the use of polynomial regression equations as an alternative to difference scores in organizational research. Academy of Management Journal, 36(6), 1577–1613.
Nestler, S., Humberg, S., & Schönbrodt, F. D. (2019). Response surface analysis with multilevel data: Illustration for the case of congruence hypotheses. Psychological Methods, 24(3), 291–308. https://doi.org/10.1037/met0000199

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2 Time-lagged correlations

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3 Interactive response surface plots

3.1 Physical health

The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.

3.2 Mental health

The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.

4 Multilevel lagged analysis results

4.1 Measurement invariance

Model fit statistics for the test of measurement invariance for vitality and learning across T1 to T4

N = 1,064. CFI = comparative fit index; TLI = Tucker–Lewis index; RMSEA = root mean square error of approximation; SRMR = standardized root mean square residual.
Model test \(\chi^2\) \(\textit{p}\) CFI TLI RMSEA SRMR \(\Delta\chi^2\) \(\Delta\)CFI \(\Delta\)TLI \(\Delta\)RMSEA \(\Delta\)SRMR
Configural invariance 404.625 (188) <.001 0.990 0.985 0.033 0.032 - - - - -
Weak invariance 412.586 (200) <.001 0.990 0.986 0.032 0.032 7.961 (12) <0.001 0.001 -0.001 <0.001
Strong invariance 434.928 (212) <.001 0.990 0.986 0.031 0.032 22.342 (12) <0.001 <0.001 <0.001 <0.001
Strict invariance 445.956 (218) <.001 0.989 0.986 0.031 0.033 11.028 (6) <0.001 <0.001 <0.001 <0.001

4.2 Lagged effects multilevel models

Lagged multilevel models including the effects of predictors measured at t-1 on health measured at t

4.2.1 Physical health

Independent variable: Vitality and learning at work (Model 2)
  Physical health predicted by learning and vitality linear, interaction, and squared terms and control variables Physical health predicted by thriving and control variables
Predictors b SE beta SE beta p b SE beta SE beta p
(Intercept) 5.42 0.17 -0.00 0.03 <0.001 5.65 0.29 -0.00 0.03 <0.001
Thriving -0.01 0.04 -0.00 0.02 0.878
Positive affect -0.01 0.04 -0.00 0.02 0.772 -0.02 0.04 -0.01 0.02 0.684
Negative affect 0.02 0.03 0.01 0.01 0.610 0.01 0.03 0.00 0.01 0.723
Demands -0.01 0.03 -0.01 0.01 0.721 -0.01 0.03 -0.01 0.01 0.644
Autonomy 0.02 0.03 0.01 0.01 0.504 0.01 0.03 0.01 0.01 0.630
Coworker support 0.00 0.03 0.00 0.02 0.932 0.01 0.03 0.00 0.02 0.858
Supervisor support 0.01 0.03 0.01 0.02 0.727 0.01 0.03 0.00 0.02 0.838
Time 0.01 0.02 0.01 0.01 0.455 0.01 0.02 0.01 0.01 0.484
Mean thriving -0.02 0.06 -0.02 0.05 0.714
Mean positive affect 0.02 0.06 0.02 0.05 0.730 -0.01 0.06 -0.01 0.06 0.834
Mean negative affect -0.31 0.04 -0.26 0.03 <0.001 -0.30 0.04 -0.26 0.03 <0.001
Mean demands -0.05 0.03 -0.05 0.03 0.131 -0.04 0.03 -0.04 0.03 0.202
Mean autonomy 0.06 0.03 0.06 0.03 0.090 0.05 0.03 0.05 0.03 0.149
Mean coworker support 0.00 0.04 0.00 0.04 0.900 -0.00 0.04 -0.00 0.04 0.943
Mean supervisor support 0.01 0.04 0.01 0.04 0.852 0.00 0.04 0.00 0.04 0.977
Age -0.01 0.00 -0.16 0.03 <0.001 -0.01 0.00 -0.17 0.03 <0.001
Gender -0.03 0.05 -0.02 0.03 0.499 -0.03 0.05 -0.02 0.03 0.492
Education 0.09 0.03 0.09 0.03 0.002 0.09 0.03 0.09 0.03 0.002
Organizational tenure 0.01 0.00 0.06 0.03 0.059 0.01 0.00 0.07 0.03 0.037
Vitality -0.01 0.03 -0.01 0.02 0.699
Learning 0.01 0.03 0.01 0.02 0.691
Vitality squared -0.05 0.03 -0.03 0.02 0.097
Vitality x learning 0.01 0.03 0.01 0.02 0.658
Learning squared 0.03 0.02 0.02 0.02 0.257
Mean vitality -0.25 0.16 -0.26 0.17 0.130
Mean learning 0.09 0.18 0.09 0.17 0.592
Mean vitality squared 0.07 0.03 0.45 0.21 0.037
Mean vitality x learning -0.03 0.05 -0.18 0.28 0.523
Mean learning squared -0.01 0.04 -0.04 0.22 0.850
Random Effects
σ2 0.275 0.265
τ00 0.400 Person.ID 0.392 Person.ID
τ11 0.011 Person.ID.Thriving 0.040 Person.ID.Vitality
ρ01 -0.087 Person.ID -0.314 Person.ID
ICC 0.593 0.604
N 888 Person.ID 888 Person.ID
Observations 2136 2136
Marginal R2 / Conditional R2 0.100 / 0.634 0.109 / 0.647

4.2.2 Mental health

Independent variable: Vitality and learning at work (Model 2)
  Mental health predicted by learning and vitality linear, interaction, and squared terms and control variables Physical health predicted by thriving and control variables
Predictors b SE beta SE beta p b SE beta SE beta p
(Intercept) 4.41 0.12 -0.00 0.02 <0.001 4.14 0.20 0.00 0.02 <0.001
Thriving -0.03 0.04 -0.01 0.02 0.490
Positive affect -0.06 0.04 -0.02 0.01 0.125 -0.04 0.04 -0.02 0.01 0.298
Negative affect 0.09 0.04 0.03 0.01 0.018 0.08 0.04 0.03 0.01 0.033
Demands -0.03 0.03 -0.01 0.01 0.397 -0.03 0.03 -0.01 0.01 0.302
Autonomy -0.02 0.03 -0.01 0.01 0.557 -0.02 0.03 -0.01 0.01 0.596
Coworker support 0.01 0.03 0.01 0.01 0.732 0.01 0.03 0.01 0.01 0.653
Supervisor support -0.01 0.03 -0.01 0.01 0.665 -0.01 0.03 -0.01 0.01 0.734
Time -0.00 0.02 -0.00 0.01 0.988 0.01 0.02 0.00 0.01 0.741
Mean thriving 0.06 0.04 0.05 0.03 0.155
Mean positive affect 0.37 0.04 0.31 0.03 <0.001 0.22 0.05 0.18 0.04 <0.001
Mean negative affect -0.66 0.03 -0.49 0.02 <0.001 -0.64 0.03 -0.48 0.02 <0.001
Mean demands -0.01 0.02 -0.01 0.02 0.542 0.00 0.02 0.00 0.02 0.951
Mean autonomy 0.02 0.02 0.02 0.02 0.326 0.04 0.02 0.03 0.02 0.101
Mean coworker support 0.06 0.03 0.05 0.03 0.033 0.05 0.03 0.04 0.02 0.091
Mean supervisor support -0.05 0.03 -0.05 0.03 0.037 -0.06 0.03 -0.06 0.03 0.013
Age 0.01 0.00 0.06 0.02 0.003 0.01 0.00 0.06 0.02 0.002
Gender -0.02 0.04 -0.01 0.02 0.557 -0.02 0.04 -0.01 0.02 0.583
Education -0.03 0.02 -0.03 0.02 0.102 -0.04 0.02 -0.03 0.02 0.079
Organizational tenure -0.00 0.00 -0.00 0.02 0.835 -0.00 0.00 -0.01 0.02 0.632
Vitality -0.08 0.04 -0.05 0.02 0.025
Learning 0.07 0.03 0.04 0.02 0.020
Vitality squared 0.00 0.03 0.00 0.02 0.946
Vitality x learning 0.01 0.03 0.01 0.02 0.697
Learning squared -0.01 0.02 -0.00 0.02 0.807
Mean vitality 0.72 0.12 0.67 0.11 <0.001
Mean learning -0.34 0.13 -0.29 0.11 0.007
Mean vitality squared -0.05 0.03 -0.26 0.14 0.065
Mean vitality x learning -0.05 0.04 -0.22 0.20 0.261
Mean learning squared 0.06 0.03 0.32 0.14 0.029
Random Effects
σ2 0.298 0.308
τ00 0.138 Person.ID 0.116 Person.ID
τ11 0.111 Person.ID.Thriving 0.038 Person.ID.Vitality
ρ01 0.066 Person.ID -0.273 Person.ID
ICC 0.342 0.294
N 888 Person.ID 888 Person.ID
Observations 2136 2136
Marginal R2 / Conditional R2 0.541 / 0.698 0.557 / 0.687